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Vassilakis, P.N. (2007b).  Representing sound energy, phase, and interference using three-dimensional signals.  J. Acoust. Soc. Am., 121(5/2): 3098 (presented at the 153nd meeting of the Acoustical Society of America, Salt Lake City, UT)
[Invited presentation - Special session on Speech Communication: Frontiers of Spectrum Analysis with Speech Applications.]

As graphic representations of vibrations/waves, sound signals capture only selected attributes of the phenomenon they represent. Assuming equivalence between signals and sound waves obscures the fact that two-dimensional signals are unfit to a) represent wave-energy quantities consistently across frequencies, b) account for the phase flip and alternating positive/negative amplitude values in modulated waves with AM-depth>100%, and c) represent the energy content of interference. The proposed sound-signal representation is based on the complex equation of motion describing a wave. It results in spiral sine signals and twisted-spiral complex signals, similar to complex analytic signals, with the imaginary component of the complex equation of motion representing the signal envelope’s argument (phase). Spiral sine signals offer a consistent measure of sine-wave energy across frequencies, while twisted-spiral complex signals account for the negative amplitudes observed in modulated signals, mapping the modulation parameters onto the twisting parameters. In terms of interference, 3-D signals illustrate that amplitude fluctuations and the signal envelopes that describe them are not just boundary curves but waves that trace changes in the total instantaneous energy of a signal over time, representing the oscillation between potential and kinetic energies within a wave. Examples of 3D animations illustrating the proposed signals are presented.